Setup

Let $X_1,X_2,\dots$ be i.i.d. random variables and let

$$ T=\inf\{n>0:X_n\in A\}. $$

Then $T$ is the first entrance time into the set $A$.

Distribution

If

$$ p=\mathbb P(X_1\in A)>0, $$

then $T$ has a geometric distribution:

$$ \mathbb P(T=k)=(1-p)^{k-1}p, \qquad k=1,2,3,\dots $$

and

$$ \mathbb E[T]=\frac1p. $$

Why it works

The event $\{T=k\}$ means:

  • the first $k-1$ trials miss $A$
  • the $k$th trial hits $A$

Independence turns this into the geometric formula.